Non-Universal Fluctuations of the Empirical Measure for Isotropic Stationary Fields on ${\mathbb{S}}^2\times \mathbb{R}$

Abstract

In this paper, we consider isotropic and stationary real Gaussian random fields defined on ${\mathbb{S}}^2\times \mathbb{R}$ and we investigate the asymptotic behavior, as $T\to +\infty$, of the empirical measure (excursion area) in ${\mathbb{S}}^2\times \mathbb{R}$ at any threshold, covering both cases when the field exhibits short and long memory, i.e. integrable and non-integrable temporal covariance. It turns out that the limiting distribution is not universal, depending both on the memory parameters and the threshold. In particular, in the long memory case a form of Berry's cancellation phenomenon occurs at zero-level, inducing phase transitions for both variance rates and limiting laws.

39 pages; comments are welcome

References in corpus (1)

• Central limit theorems for sequences of multiple stochastic integrals